A stone projected from ground with certain speed at an angleθ with the horizontal attains max height h1 when - Physics - Motion in a plane

Question

A stone projected from ground with certain speed at an
angleθ with the horizontal attains max height h1 when it is
projected with same speed at an angleθ with the vertical
attains height H2 the horizontal range will be

Satyendra Singh · Accepted Answer

Say the stone is projected with a velocity of 'v' in both the cases
then in the first case:

also, h1 = v2 sin2 θ2g or v2 = 2h1gsin2 θ ------(1)

and,

h2 = v2 (sin (90-θ))22g or v2 = 2h2gcos2 θ ------(2)

Range1=v2 sin2θgalso, since projection in the secon case took
place at an angle θ from the vertical it means it took place
at an anlge (90-θ) from the horizontal:therefore, Range2 = v2
sin 2(90-θ)g = v2 sin(180-2θ)g = v2 sin2θg
-------(3)or, Range1 = Range2 = v2 sin2θg

Now, since
Range1 = Range2 = 2h1g sin2θsin2 θ (after substituting
the value of v2 from (1)) =2h1g 2sinθcosθsin2 θ =
4h1g cosθsin θ = 4h1g cot θ ------(Ans(1))also, if
we substitute the value of v2 from (2), then,Range1 = Range2 = 2h2g
sin2θcos2 θ = 4h2g sinθcos θ = 4h2g tanθ
--------(Ans (2))Both Ans(1) and Ans (2) are viable answers

A stone projected from ground with certain speed at an angleθ with the horizontal attains max height h1. when it is projected with same speed at an angleθ with the vertical attains height H2. the horizontal range will be